Download The Interaction of Radiation and Matter: Semiclassical Theory (cont

The Interaction of Radiation and Matter:
Semiclassical Theory (cont.)
II. Review of Basic Quantum Mechanics: Dynamic
Behavior of Quantum Systems: (pdf copy)
Quantum Mechanical Equations of Motion:
See the Dirac Notation Sheet
To this point in our review, we have been concerned with describing the states of a
system at one instant of time. The complete dynamical theory must describe, of course,
connections between different instants of time.
"When one makes an observation on the dynamical system, the state of the system gets
changed in an unpredictable way, but in between observations causality applies, in
quantum mechanics as in classical mechanics, and the system is governed by equations
of motion which make the state at one time determined the state at a later time." [1]
Thus, it is only the disturbance caused by the interaction of a system with a measuring
device that makes the system's behavior cease to be strictly causal.
Schrödinger Equation of Motion:
Consider the time evolution of a particular state of an undisturbed system. To deal with
such a dynamical system, we need a linear operator of the form[2]
[ II-1 ]
Passing to the limit
, this operator yields a related linear operator for the time
derivative of the state vector with respect to
[ II-2a ]
If it is postulated that
where
is the total Hamiltonian (energy) of
the system,[3] we obtain the Schrödinger equation of motion in abstract form -- viz.
[ II-2b ]
Equations [ II-1 ] and [ II-2b ] are consistent if
.
[ II-3 ]
In the Schrödinger representation, the abstract Schrödinger equation of motion
becomes[4]
[ II-4 ]
Heisenberg Equation of Motion:
In the Schrödinger picture, as outlined above, we picture the states of the undisturbed
motion by associating each state with a moving ket, the state at any time corresponding
to the ket at that time. In the Heisenberg picture a unitary transformation is applied
which brings to rest the kets corresponding to states of undisturbed motion. In this
picture, the appropriate equation of motion is one describing the motion or evolution of
linear operators (dynamic variables) -- viz.
or
[ II-5a ]
where
is a fixed Schrödinger dynamic variable and
is a time varying
Heisenberg dynamic variable. Differentiating with respect to time
[ II-5b ]
and then using Equation [ II-3 ], we obtain the Heisenberg form of the equation of
motion as
[ II-6 ]
where
. Note that this equation -- i.e. Equation [ II-6 ] – is consistent
with the classical analogy discussed earlier.
Energy Eigenvector Representation -- Heisenberg Representation
In the Heisenberg picture the stationary states
a time independent Hamiltonian .
correspond to fixed eigenvectors of
CASE 1 -- Time independent Hamiltonian:
To study the time evolution of a given state when the Hamiltonian is time independent,
we expand the state vector in terms of these fixed energy eigenvectors and write
Equation [ II-2b ] as
[ II-7a ]
and
[ II-7b ]
Therefore
[ II-8 ]
We may expand an arbitrary wave function in terms of these eigenvectors -- viz.
[ II-9 ]
where
and the coefficients
course independent of time.
are, in this instance, of
CASE 2 -- Time dependent Hamiltonian:
Consider now a time dependent Hamiltonian in the form
[ II-10 ]
In this instance, we may write Equation [ II-2b ] as
[ II-11a ]
which becomes, on operating from the left with
[ II-11b ]
If we write
-- see Equation [ II-8 ] -- then
[ II-11c ]
Therefore, when we expand an arbitrary wave function a la Equation [ II-9 ], we find
[
II12a
]
and
[ II-12b ]
where
and
[ II-13 ]
If we write
,
obeys the equation of motion
[ II-14 ]
Time Dependent Perturbation Theory:
Elementary Ideas -- First Order Iteration:
In the set of equations denoted as Equation [ II-12b ] assume that
[ II-15 ]
so that in first-order
[ II-16 ]
Suppose that
[ II-17 ]
for
, then
[ II-18 ]
CASE 1:
[ II-19 ]
which is valid as long as
. Equation [ II-18 ] shows that transitions are more
likely if energy is conserved between initial and final states.
CASE 2:
In the so called rotating-wave approximation we neglect the first term in Equation [ II18 ] so that
[ II-20 ]
which is again valid as long as
. This equation shows that transitions are
unlikely unless the resonance condition
is satisfied.
Let us suppose that we have continuum of energy levels represented by a density of
states function
. Therefore the total probability for transition out of the initial state
has the approximate value
[ II-21a ]
For
[ II-21b ]
so that initial the transition rate -- i.e.,
-- is linearly dependent on . For longer
times, it is reasonable to assume that the "frequency-width" of
is large
compared to the inverse of the elapsed time -- or more precisely
approximate the total transition probability as
--we may further
[ II-21c ]
which yields the famous Fermi Golden Rule for the transition rate
[ II-22 ]
Higher-Order Time Dependent Perturbation Theory:
Consider once again perturbation solutions for a Hamiltonian of the
form
. At outset we transform our equations to the halfway land
of the interaction picture by first rewriting Equation [ II-2b ] to emphasize that it deals
with a state vector in the Schrödinger picture -- viz.
[ II-2b' ]
Then we make the unitary transformation
[ II-23 ]
so that Equation [ II-2b' ] becomes
[ II24 ]
Using a version of Equation [ II-3 ] we obtain the equation of motion for the state in the
interaction picture -- viz.
[ II-25 ]
If we write
[ II-26a ]
so that
[ II-26b ]
We may formally integrate this equation to obtain
[ II-27a ]
By iterating once we have
[ II-27b ]
or by successive iterations we have to an arbitrary level of precision
[ II-27c ]
Density Operator (Matrix):
In classical theory any state of a dynamic system is represented as a point in the phase
space whose number of dimensions is twice (coordinate and momentum) the number of
degrees of freedom in the system. This point will move according to the classical
equations of motion. If the state of the system is defined by some probabilistic
specification, then we know only the probability that the system may be assigned a
given phase point at a particular time. We may envisage the time varying probabilistic
specification as a fluid of density
moving through phase space.
Each particle of the fluid will move according to the system's equations of motion. The
following conservation relationship holds:
[ II-28 ]
John von Neumann first introduced a corresponding density function into quantum
mechanics. Dirac pointed out that the existence of such a quantum mechanical density is
surprising in view of the fact that phase space has no meaning in quantum mechanics
since numerical values cannot be assigned simultaneously to the coordinates and
momenta. The quantum mechanical density operator (dyadic) is defined as
[ II-29 ]
where
is the probability of the system being in the state
.
The equation of motion for the density operator is easily determined from the
Schrödinger equation -- i.e. Equation [ II-2b ] -- as
[ II-30 ]
.
This equation is, thus, the classical analogy of Equation [ II-28 ]. Let us now express the
expectation value of an operator
in terms of the density operator -- viz.
[ II-31 ]
Gibbs showed that when a dynamic system is in thermodynamic equilibrium with its
surroundings at a given temperature T, the phase space density is given by
.
[ II-32a ]
This formula is taken over unchanged into quantum mechanics.
[ II-32b ]
APPENDIX - TIME INDEPENDENT PERTURBATION THEORY
Suppose that we have a Hamiltonian
where is a big part and
is a
small part. Of course, we want the solution to the complete eigenvalue problem -- i.e.,
[ A-1 ]
To obtain an approximate solution we expand the eigenvalues and eigenvectors in a
series of terms of increasing smallness - viz.
[ A-2a ]
[ A-2b ]
[ A-2c ]
Let us first consider normalization - i.e.
[ A-3a ]
Compare terms of equal smallness so that
[ A-3b ]
[ A-3c ]
[ A-3d ]
etc.
From Equation [ A-3c ] we see that we may take
without introducing
error. Next we write an expanded version of the eigenvalue equation [ A-1 ]
[ A-4a ]
Again compare terms of equal smallness so that
]
[ A-4b
[ A-4c ]
[ A-4d ]
etc.
Operate through Equation [ A-4c ] with
and [ A-4b ] we find
and using Equations [ A-3b ]
[ A-5 ]
Operate through Equation [ A-4c ] with
and using again Equations [ A-3b ]
and [ A-4b ] we find
[ A-6a ]
or we have the representative
[ A-6b ]
and
[ A-6c ]
Operate through Equation [ A-4d ] with
and [ A-4b ] we find
and, of course, using Equations [ A-3b ]
[ A-7a ]
or
[ A-7b ]
[1] From Section 27 of P. A. M. Dirac, The Principles of Quantum Mechanics (Revised
fourth edition), Oxford University Press (1967).
[2] This is first member of a class of "displacement " operators that we may treat in a
similar fashion. See Section 25 of P. A. M. Dirac, The Principles of Quantum
Mechanics (Revised fourth edition), Oxford University Press (1967).
[3] There are two justifications of this postulate: (a) analogy with classical mechanics
(see Equation [ II-6 ]) and (b) relativistic invariance vis-a-vis Equation [ I-20 ].
[4] Since
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This page was prepared and is maintained by R. Victor Jones, [email protected]
Last updated March 3, 2000